Repeating a proof from the literature in my paper

Question

This question is about writing a paper in mathematics.

I am studying a (partial differential) equation and an algorithm to solve it approximately (finite element method). I need to report in my work a theorem on the approximation error of this algorithm. Such theorem already exists in the paper X: it is proved in a context that is more general than mine for some reasons (moving vs non-moving domain), and slightly less general than mine for other reasons (homogeneous vs non-homogeneous boundary conditions).

Should I report the adapted version of the proof from paper X, after of course appropriately citing it?

The modifications I had to do to the existing proof are mostly "trivial": my estimations are just longer and slightly more technically involved versions of those in X. In one point, however, I make a non-trivial (not so ingenious either) conceptual passage. I need a lot of context to report and prove this conceptual passage, so that rewriting the whole proof in my paper is not so much additional work, and it would be of course clearer.

Long story short, reporting the whole proof seems pedantic and may not be well received from the reviewers, whereas saying "The proof is as in [X]." feels lazy or not informative enough to me. But there may be more leftover space for reporting original content directly related to my publication!

Any hints?

Answer 1

Synthesizing a few answer comments as well as a suggestion of my own:

• Ask your adviser if you're a student. They're likely to have a better gut feeling on both the specific result and community expectations.
• If you're writing up, probably not the worst things in the world to write up the proof, finish drafting the paper, then consider whether your target journal or similar papers have explicit or implicit length guidelines. Then know you can remove the proof if the draft's too long.
• If the proof is essentially routine, a bit longish, but has a nontrivial element: Consider deferring it to an Appendix. This makes it extremely easy to cut if a reviewer doesn't think it's necessary.

Answer 2

Your concern about it being pedantic and not well received is unfounded. It is a healthy attitude that since their result does not cover the case you need, you consider it your (unfortunate) duty to give a proof. It may be pedantic, but pedantic in a good way, nobody should reproach you for it.

In my opinion, "the proof is as in X" should be considered an option only if the proof is verbatim as in X. Even then, it's better to acknowledge something like "the result in X formally does not apply to our case, but the proof transfers verbatim." If the proof is applies mutatis mutandis, write "the proof works verbatim once one replaces all derivatives with finite differences." In other words, it is an option to describe a proof in a "perturbative way" - by referring to an existing proof in the literature and describing what has to be changed. But at some point it may get awkward, and it becomes cleaner to give a self-contained exposition. It's your call as an author.

It's good, of course, if the role of the proof is properly explained: "We follow closely [1], with modification that may be obvious for an experienced reader. Nevertheless, because (reasons), we chose to include the proof." Relegating the proof to appendix, as suggested, is also a good way to signal your view of its originality.

If you submit to a journal with a page limit and have to cut out the proof, it would be a service to the community to keep a longer version publicly available on arXiv or your webpage.